As a method of drawing an acrylic fiber bundle, drawing in a pressurized steam atmosphere is known. This is because temperature higher than hot water in an atmospheric pressure can be obtained and moisture existing therein causes an effect of plasticizing the acrylic fiber bundle so that the fiber bundle can be drawn up to a high magnification. The carbon-fiber precursor acrylic fiber bundle used as a precursor of the carbon fiber particularly does not have a noticeable melting point due to the high acrylonitrile content, and the fiber bundle substantially cannot be drawn only by the effect of the high temperature. For this reason, the pressurized steam drawing process is used in many cases. However, when the acrylic fiber bundle is drawn up to a high magnification by the steam drawing process, problems in which single filaments are broken, fluffs occur, or the entire fiber bundle is broken may occur. The same applies to the cases where a fine fiber is to be obtained or the process is to be performed at a high speed.